By Prof. Alberto Cambini, Prof. Laura Martein (auth.)

The authors have written a rigorous but trouble-free and self-contained booklet to give, in a unified framework, generalized convex features, that are the numerous non-convex features that percentage at the least one of many beneficial homes of convex services and that are usually more advantageous for describing real-world difficulties. The e-book might be a great tool not just for researchers, but in addition for graduates and complicated scholars operating in economics, mathematical programming, the administration sciences and operations examine. It starts with a overview of convex research and the basic theoretical findings on generalized convexity and on optimization, together with their functions. The textual content maintains with an introductory bankruptcy dedicated to generalized monotonicity and its dating to generalized convexity, with the characterizations of significant periods of fractional programming, and with theoretical houses and sequential equipment. The publication additionally contains various routines and appendices which checklist the findings consulted.

"I can hugely suggest this glorious new quantity to somebody who's attempting to train on generalized convexity/generalized monotonicity in an utilized arithmetic division or in a qualified tuition. the quantity comprises proofs and routines. It additionally presents adequate references when you are looking to dig deeper as graduate scholars and as researchers. With commitment and masses love the authors have produced a publication that's obtainable for somebody with a constrained history in easy arithmetic on the collage point. even as, it additionally ends up in extra refined arithmetic should you spend extra time on a specific chapter."

From the Foreword, Siegfried Schaible, collage of California, Riverside, USA

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**Additional info for Generalized convexity and optimization: theory and applications**

**Sample text**

1. 35. Prove that a function of one variable is homogeneous of degree α if and only if it is of the form f (x) = kxα . 36. Let f be a linearly homogeneous function deﬁned on a convex cone C ⊆ n . Prove that f is concave if and only if f (x + y) ≥ f (x) + f (y) for every x, y ∈ C. 37. A relevant result regarding homogeneous functions is Euler’s Theorem: let f be a diﬀerentiable function deﬁned on the open convex cone C ⊆ n . Then, f is homogeneous of degree α if and only if xT ∇f (x) = αf (x), ∀x ∈ C.

9. Let f be a continuous diﬀerentiable function on an open convex set S ⊆ n . Then, f is pseudoconvex (strictly pseudoconvex) on S if and only if the following conditions hold: (i) f is quasiconvex on S; (ii) If x0 ∈ S, ∇f (x0 ) = 0, then x0 is a local minimum (strict local minimum) for f . Proof. 5. Assume now that (i) and (ii) hold. 8 we must prove that if x0 ∈ S and u ∈ n are such that uT ∇f (x0 ) = 0, the function ϕ(t) = f (x0 + tu) attains a local minimum at t = 0. If ∇f (x0 ) = 0, the thesis follows from (ii); if ∇f (x0 ) = 0, the continuity of the gradient map implies the existence of an open neighbourhood U (x0 ) of x0 such that ∇f (x0 ) = 0, ∀x ∈ U (x0 ).

Consider the restriction ϕ(t) = f (x1 + t(x2 − x1 )), t ∈ [0, 1]. 2) holds. 2) holds. 1, there exist t1 , t2 ∈ [0, 1] such that ϕ(t1 ) ≥ ϕ(t2 ) and ϕ (t1 )(t2 − t1 ) > 0. Set x ¯1 = x1 + t1 (x2 − x1 ), x ¯2 = x1 + t2 (x2 − x1 ); we have f (¯ x1 ) ≥ f (¯ x2 ) 1 T and ϕ (t1 ) = (x2 − x1 )T ∇f (¯ x1 ) = t2 −t (¯ x − x ¯ ) ∇f (¯ x ). 2). 2) we also have a strict inequality on the right-hand side when S is open and x1 is not a critical point. 1 The diﬀerentiability of a function g on a set X ⊆ on an open set containing X.